\(A=\frac{sin3x+sinx}{2cosx}=\frac{2sin2x.cosx}{2cosx}=sin2x\)

`A=[sin x+sin 2x+sin 3x]/[cos x+cos 2x+cos 3x]`

`A=[(sin x+sin 3x)+sin 2x]/[(cos x+cos 3x)+cos 2x]`

`A=[2sin 2x.cos (-x)+sin 2x]/[2cos 2x.cos (-x)+cos 2x]`

`A=[sin 2x(2cos(-x)+1)]/[cos 2x(2cos(-x)+1)]`

`A=[sin 2x]/[cos 2x]=tan 2x`.

`A=[sin x + sin 2x + sin 3x]/[cos x + cos 2x + cos 3x]`

`A=[2sin2x.cosx+sin2x]/[2cos2x.cosx+cos2x]`

`A=[sin2x(2cosx+1)]/[cos2x(2cosx+1)]`

`A=tan 2x`

\(A=\dfrac{sinx-sin2x+sin3x}{cosx-cos2x+cos3x}\)

\(ĐK\left\{{}\begin{matrix}cos2x\ne0\\cosx\ne\dfrac{1}{2}\end{matrix}\right.\)  \(\Leftrightarrow\)  \(A=\dfrac{sinx+sin3x-sin2x}{cosx+cos3x-cos2x}\)     

\(\Leftrightarrow\)  \(\left\{{}\begin{matrix}=\dfrac{2sin2x.cosx-sin2x}{2cos2x.cosx-cos2x}\\=\dfrac{sin2x\left(2cosx-1\right)}{cos2x\left(2cosx-1\right)}\end{matrix}\right.\)  \(\Rightarrow\) \(A=tan2x\)

ĐKXĐ:

\(sin3x-sinx\ne0\)

\(\Leftrightarrow sin3x\ne sinx\)

\(\Leftrightarrow\left\{{}\begin{matrix}3x\ne x+k2\pi\\3x\ne\pi-x+n2\pi\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}x\ne k\pi\\x\ne\frac{\pi}{4}+\frac{n\pi}{2}\end{matrix}\right.\)

TXĐ: \(sin3x-sinx\ne0\)

\(\Leftrightarrow sin3x\ne sinx\)

\(\Leftrightarrow\left[{}\begin{matrix}3x\ne x+k2\pi\\3x\ne\pi-x+k2\pi\end{matrix}\right.\)

\(\Rightarrow\left[{}\begin{matrix}x\ne k\pi\\x\ne\frac{\pi}{4}+\frac{k\pi}{2}\end{matrix}\right.\)

\(P=\dfrac{sin2x+sinx}{\dfrac{1}{2}\cdot cosx\cdot sin2x+sin2x}=\dfrac{sinx\left(2cosx+1\right)}{sin2x\left(\dfrac{1}{2}cosx+1\right)}\)

\(=\dfrac{2cosx+1}{2\cdot cosx\cdot\left(\dfrac{1}{2}cosx+1\right)}\)

a) \(ĐKXĐ:\hept{\begin{cases}x\ne0\\x\ne-1\end{cases}}\)

\(M=\left(\frac{x+2}{3x}+\frac{2}{x+1}-3\right):\frac{2-4x}{x+1}-\frac{3x-x^2+1}{3x}\)

\(=\left[\frac{\left(x+2\right)\left(x+1\right)}{3x\left(x+1\right)}+\frac{6x}{3x\left(x+1\right)}-\frac{9x\left(x+1\right)}{3x\left(x+1\right)}\right].\frac{x+1}{2-4x}+\frac{x^2-3x-1}{3x}\)

\(=\left[\frac{x^2+3x+2}{3x\left(x+1\right)}+\frac{6x}{3x\left(x+1\right)}-\frac{9x^2+9x}{3x\left(x+1\right)}\right].\frac{x+1}{2-4x}+\frac{x^2-3x-1}{3x}\)

\(=\frac{x^2+3x+2+6x-9x^2-9x}{3x\left(x+1\right)}.\frac{x+1}{2-4x}+\frac{x^2-3x-1}{3x}\)

\(=\frac{2-8x^2}{3x}.\frac{1}{2\left(1-2x\right)}+\frac{x^2-3x-1}{3x}\)

\(=\frac{2\left(1-4x^2\right)}{3x}.\frac{1}{2\left(1-2x\right)}+\frac{x^2-3x-1}{3x}\)

\(=\frac{2\left(1-2x\right)\left(1+2x\right)}{3x}.\frac{1}{2\left(1-2x\right)}+\frac{x^2-3x-1}{3x}\)

\(=\frac{1+2x}{3x}+\frac{x^2-3x-1}{3x}\)

\(=\frac{1+2x+x^2-3x-1}{3x}=\frac{x^2-x}{3x}=\frac{x\left(x-1\right)}{3x}=\frac{x-1}{3}\)

b) Với \(x=6013\)( thỏa mãn ĐKXĐ )

Thay \(x=6013\)vào biểu thức ta được: 

\(M=\frac{6013-1}{3}=\frac{6012}{3}=2004\)

\(A=\frac{cosx-cos3x+cos4x-cos2x}{sinx-sin3x+sin4x-sin2x}=\frac{2sin2x.sinx-2sin3x.sinx}{-2cos2x.sinx+2cos3x.sinx}\)

\(=\frac{sin2x-sin3x}{cos3x-cos2x}=\frac{-2cos\left(\frac{5x}{2}\right)sin\left(\frac{x}{2}\right)}{-2sin\left(\frac{5x}{2}\right)sin\left(\frac{x}{2}\right)}=cot\left(\frac{5x}{2}\right)\)

\(B=sinx+2cos2x.sinx+2cos4x.sinx+2cos6x.sinx\)

\(=sinx+sin3x-sinx+sin5x-sin3x+sin7x-sin5x\)

\(=sin7x\)